Form the Jacobi theta exponents without amplified rounding

doc/sf/jacobi_theta.qbk

@@ -55,6 +55,10 @@ A more accurate computation will take advantage of [tau]:
 
 Internally, when /q/ is larger than exp(-[pi]) (that is, when [tau] is less than 1), Boost implements the /q/ parameterization by taking the logarithm of /q/ and passing it to the [tau] parameterization; as such, using the [tau] parameterization directly will generally yield greater precision in that regime.
 When /q/ is smaller than exp(-[pi]), the Fourier series is summed directly in terms of /q/, and the two parameterizations are equally accurate.
+
+Every term of the series is an exponential whose argument is a product or quotient of [tau], [pi] and a small integer, or, in the transformed series used when [tau] < 1, the square of /z/ plus a multiple of [pi]/2.
+Rounding such an argument to working precision would cost as many ulps in the term as the argument is large, so the rounding errors of [pi], of the products, of the division and of the reduction of /z/ are tracked exactly (using error-free transformations that work for any binary floating-point type) and applied as a correction factor to each term.
+The [tau] parameterization is therefore accurate to a few ulps throughout, apart from the conditioning of the function itself. In the /q/ parameterization the rounding of log(/q/) remains, and its effect grows as /q/ approaches 1.
 As another example, if the complement of /q/ is known with great accuracy, then instead of:
 
   jacobi_theta1(x, 1-q_complement);
@@ -125,7 +129,7 @@ Fixing /x/=5 and varying /q/, the ULPs plot looks like:
 
 [graph jacobi_theta1q_float]
 
-Accuracy tends to degenerate near /q/=1 (small [tau]).
+Accuracy degrades gently as /q/ approaches 1 (small [tau]), where the rounding of log(/q/) is amplified by the size of the exponents in the transformed series; the [tau] parameterization does not have this problem.
 
 [heading Implementation]
 
@@ -194,7 +198,7 @@ Fixing /x/=0.4 and varying /q/, the ULPs plot looks like:
 
 [graph jacobi_theta2q_float]
 
-Accuracy tends to degenerate near /q/=1 (small [tau]).
+Accuracy degrades gently as /q/ approaches 1 (small [tau]), where the rounding of log(/q/) is amplified by the size of the exponents in the transformed series; the [tau] parameterization does not have this problem.
 
 [heading Implementation]
 
@@ -276,7 +280,7 @@ Fixing /x/=0.4 and varying /q/, the ULPs plot looks like:
 
 [graph jacobi_theta3q_float]
 
-Accuracy tends to degenerate near /q/=1 (small [tau]).
+Accuracy degrades gently as /q/ approaches 1 (small [tau]), where the rounding of log(/q/) is amplified by the size of the exponents in the transformed series; the [tau] parameterization does not have this problem.
 
 [heading Implementation]
 
@@ -358,7 +362,7 @@ Fixing /x/=5 and varying /q/, the ULPs plot looks like:
 
 [graph jacobi_theta4q_float]
 
-Accuracy tends to degenerate near /q/=1 (small [tau]).
+Accuracy degrades gently as /q/ approaches 1 (small [tau]), where the rounding of log(/q/) is amplified by the size of the exponents in the transformed series; the [tau] parameterization does not have this problem.
 
 [heading Implementation]